Penetration depth of an electric field in a semi-infinite plasma

It is shown that the penetration of an oscillating electric field in a semi-infinite classical plasma obeys the standard exponential attenuation law $e^{-x/\lambda_{e}}$ (besides oscillations), where $x$ is the distance from the wall and $\lambda_{e}$ is the extinction length (penetration depth, attenuation length). The penetration depth is computed here explicitly; it is shown that it is of the order $\lambda_e \simeq [\mid\varepsilon \mid/(1-\varepsilon)]^{1/3}v_{th} / \omega$, where $\varepsilon$ is the dielectric function, $\omega$ is the frequency of the field and $v_{th}=\sqrt{T/m}$ is the thermal velocity ($T$ being the temperature and $m$ the particle (electron) mass). The result is obtained by including explicitly the contribution of the surface term.